Energy Conservation for the Compressible Euler and Navier-Stokes Equations with Vacuum

TL;DR

This paper establishes new sufficient conditions under which weak solutions to the compressible Euler and Navier-Stokes equations conserve energy, even in the presence of vacuum regions, by relaxing regularity assumptions on pressure laws.

Contribution

It introduces novel criteria for energy conservation in compressible flows with vacuum, extending previous results to more realistic pressure laws and degenerate viscosities.

Findings

Energy conservation holds under divergence-measure velocity fields.

Additional integrability conditions near vacuum ensure energy conservation.

Results extend to bounded domains and Navier-Stokes equations with degenerate viscosity.

Abstract

We consider the compressible isentropic Euler equations on $\mathbb{T}^d\times [0,T]$ with a pressure law $p\in C^{1,\gamma-1}$, where $1\le \gamma <2$. This includes all physically relevant cases, e.g.\ the monoatomic gas. We investigate under what conditions on its regularity a weak solution conserves the energy. Previous results have crucially assumed that $p\in C^2$ in the range of the density, however, for realistic pressure laws this means that we must exclude the vacuum case. Here we improve these results by giving a number of sufficient conditions for the conservation of energy, even for solutions that may exhibit vacuum: Firstly, by assuming the velocity to be a divergence-measure field; secondly, imposing extra integrability on $1/\rho$ near a vacuum; thirdly, assuming $\rho$ to be quasi-nearly subharmonic near a vacuum; and finally, by assuming that $u$ and $\rho$ are H\"older continuous. We then extend these results to show global energy conservation for the domain $\Omega\times [0,T]$ where $\Omega$ is bounded with a $C^2$ boundary. We show that we can extend these results to the compressible Navier-Stokes equations, even with degenerate viscosity.